* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(p) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [14] x2 + [0]
p(minus) = [1] x1 + [0]
p(p) = [1] x1 + [5]
p(s) = [1] x1 + [2]
Following rules are strictly oriented:
div(0(),s(Y)) = [14] Y + [28]
> [0]
= 0()
p(s(X)) = [1] X + [7]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
div(s(X),s(Y)) = [1] X + [14] Y + [30]
>= [1] X + [14] Y + [30]
= s(div(minus(X,Y),s(Y)))
minus(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
minus(s(X),s(Y)) = [1] X + [2]
>= [1] X + [5]
= p(minus(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
- Weak TRS:
div(0(),s(Y)) -> 0()
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(p) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [8]
p(div) = [1] x1 + [1] x2 + [8]
p(minus) = [1] x1 + [1]
p(p) = [1] x1 + [8]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
minus(X,0()) = [1] X + [1]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [1] Y + [16]
>= [8]
= 0()
div(s(X),s(Y)) = [1] X + [1] Y + [8]
>= [1] X + [1] Y + [9]
= s(div(minus(X,Y),s(Y)))
minus(s(X),s(Y)) = [1] X + [1]
>= [1] X + [9]
= p(minus(X,Y))
p(s(X)) = [1] X + [8]
>= [1] X + [0]
= X
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(s(X),s(Y)) -> p(minus(X,Y))
- Weak TRS:
div(0(),s(Y)) -> 0()
minus(X,0()) -> X
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(div) = {1},
uargs(p) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{div,minus,p}
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [2] x_1 + [1] x_2 + [5]
p(minus) = [1] x_1 + [3]
p(p) = [1] x_1 + [1]
p(s) = [1] x_1 + [8]
Following rules are strictly oriented:
div(s(X),s(Y)) = [2] X + [1] Y + [29]
> [2] X + [1] Y + [27]
= s(div(minus(X,Y),s(Y)))
minus(s(X),s(Y)) = [1] X + [11]
> [1] X + [4]
= p(minus(X,Y))
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [1] Y + [13]
>= [0]
= 0()
minus(X,0()) = [1] X + [3]
>= [1] X + [0]
= X
p(s(X)) = [1] X + [9]
>= [1] X + [0]
= X
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))